Forty-Two: The Accidental Answer to Everything
Douglas Adams was a British author and humorist who made his debut on BBC radio writing scripts for humorous radio shows.
His most famous script was eventually adapted by him to a book, then a BBC television series, then many more books, and then a movie after his death. The genius of Adams’ humor was entirely through his brilliant use of the English language. Should I describe the plot of a few of Adams’ books to you, and they would sound very silly and juvenile. However, if you read them in Adams’ own words, it is the way in which he wrote which made them particularly hilarious.
This article relates specifically to a mathematical mystery Adams just happened across without any intention of doing so. It is a mystery well known to anyone who traffics in what we call “Geek Culture,” but there is more to it than frequently meets the eye.
The Hitchhiker’s Guide to the Galaxy
The book for which Adams is most well known is a science fiction comedy which involves an intentionally complex plot that roves all across the galaxy and across eons of history, including the creation of the planet Earth.
A central point in this plot is the construction of a mind-bogglingly sophisticated computer and artificial intelligence known as “Deep Thought.” When the final touch on this computer was completed, its inventors approached it with a single question. “What,” they asked, “Is the answer to the ultimate question of life, the universe, and everything.” The machine pondered this question for a moment, and then told the inventors that it would take it 7.5 million years to calculate the answer.
So the inventors died without knowing the answer and passed the mission on to their progeny in the far, far distant future. After a waiting 300 thousand generations, the distant ancestors of the original creators approached the computer to find the answer. In one of the most comical moments in the entire book, the computer gravely announced that the answer to the ultimate question of life, the universe, and everything was “42.”
This was, of course, a joke. A simple throwaway line dreamed up by Adams to amuse the reader and advance the plot. Nevertheless it has become something of an obsession for Adams’ readers ever since, with people everywhere trying to decode some deeper meaning or significance to the number. When interviewed on the topic, and aware of the decades of obsession it had spawned, Adams stated:
It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one. I sat at my desk, stared into the garden and thought ’42 will do.’ I typed it out. End of story.
But this inconsequential decision on the part of Adams has become culturally significant. For instance, if you ask your closest web browser or A.I. “what is the ultimate answer to life the universe and everything” very likely it will respond with “42.”
The Base-13 Red Herring
Near the end of the second book, The Restaurant at the End of the Universe, Arthur Dent tries to coax the lost Ultimate Question out of his own subconscious by drawing letter tiles at random from a bag. They spell out: “What do you get if you multiply six by nine?”
Six times nine, of course, is 54. That is the joke. The question that was supposed to make sense of everything is simple arithmetic, and it is wrong. Fans soon noticed, however, that in base 13 the number 54 is written “42,” since four thirteens plus two make 54. It is a delightful coincidence, and Adams treated it as exactly that. Asked about it, he replied, “I may be a sorry case, but I don’t write jokes in base 13.”
Binary has fared better with the fans. In base 2, 42 is written 101010, which is why some fans threw parties on October 10, 2010. Written as a date, 10/10/10 is 42 in binary.
The Search for Meaning
Is it possible to give this number some greater meaning? The short answer is “sort of.” Sadly for our Adams fans, you could take literally any number and mine some significance out of it. Take, for instance, the movie titled The Number 23. In the movie, the protagonist finds a book titled The Number 23, obsesses over its contents, and finds a variety of patterns between that number and his personal life. Numbers are like that. Especially smallish ones.
Nevertheless, it is worthwhile seeing the many meanings that obsessive Adams fans have unearthed over the last half-century since the book was published. Let us delve into the deeper meaning behind the answer to Life, the Universe, and Everything.
42 In Ancient History and Culture
The number 42 is recurrent throughout ancient Egyptian culture. The corpse of the god Osiris is divided into 42 pieces. The pieces, when located, are buried under 42 shrines, and during the judgment of souls in the afterlife, the dead had to declare before the 42 judges that they had not committed any of 42 sins.
In all likelihood, the fact that there were forty-two shrines was a coincidence related to where centers of worship sprung up in Ancient Egypt, and the mythology related to that number was established to explain and lend significance to the existing shrines, which just happened to be 42 in number. In other words, 42 emerged organically from geography and culture rather than some deeper symbolic meaning.
Nevertheless, this coincidence of Egyptian geography may have carried forward through tradition to give the number more significance. And if you wanted to get further into the weeds, it might be worth investigating what it is about Egyptian history and geography which led to the construction of these original 42 shrines and the legends which thereafter emerged.
We won’t do that, however. We’ll jump forward a bid in history and look at Ancient Greece. One of the bits of Greek mythology which may very well have been related to historical fact was the journey of Pheidippides between the battlefields of Marathon and the city of Athens. In this story, the soldier Pheidippides watches in pride as his forces finally defeat the invading Persians. His pride turns to horror as he happens to see a Persian boat launch into the sea in the direction of Athens. Pheidippides realizes that the Persians are likely going to arrive in Athens and falsely inform the government that they, the Persians, had won the war — forcing Greece to sign a concession.
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To prevent this disaster, Pheidippides sheds his weapons, armor, and finally his clothes as he makes a mad dash across land toward Athens to declare that it was Greece, not Persia, who had won the day.
According to legend, he arrived at the Senate in Athens, naked and exhausted after running without stopping, and managed to give them the good news before collapsing dead.
How does this story relate to our fabled number? Well it just so happens that the marathon distance is 42.2 kilometers. The significance of this becomes somewhat less impressive when one realizes that a kilometer is a relative measure of distance. In miles, a marathon is 26.2. Or, as the ancient Greeks would have measured it, a marathon is 267.6 stadia.
More to the point, the distance does not come from ancient Greece at all. The first Olympic marathon in 1896 was about 40 kilometers. The 42.195-kilometer figure comes from the course laid out for the 1908 London Olympics, which ran from Windsor Castle to the stadium at White City, and it was adopted as the official standard only in 1921.
A little more concrete to our purposes is the historical fact that ancient Tibet had exactly 42 rulers, or that the first book ever printed by the Gutenberg printing press had 42 lines of text per column.
Not impressed with its historical significance? Perhaps it is time to delve into the mathematical significance of this enigmatic number.
The Number 42 in Mathematics
Perhaps history is the wrong place to look. Mathematicians have their own ways of making a number interesting, and 42 does turn up in a few respectable places:
· It is a sum of odd powers of two. 2 + 8 + 32 = 42, or 2¹ +2³ +2⁵. Adding up the first few odd powers of two gives the sequence 2, 10, 42, 170, 682… In binary, every member is just “10” repeated, and members become rarer and rarer as the numbers grow.
· It is a sum of powers of six. 6 + 36 = 42. Keep going (6, 42, 258, 1554…) and you get another sequence that thins out quickly.
· It is a Catalan number. The Catalan numbers (1, 1, 2, 5, 14, 42, 132…) answer a surprising variety of counting questions. Leonhard Euler met them while asking how many ways a convex polygon can be cut into triangles using straight lines between its corners: a seven-sided polygon can be divided in exactly 42 ways. The sequence later took the name of the Belgian-born mathematician Eugène Catalan (1814–1894), who connected it to the number of ways pairs of parentheses can be correctly nested. Catalan numbers grow explosively. Only fourteen of them are smaller than one million.
· It is a “practical” number. Every whole number from 1 to 42 can be built by adding together some of 42’s distinct divisors (1, 2, 3, 6, 7, 14, 21, and 42). Other practical numbers include 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, and 40, and no simple formula is known that generates them.
All of this is fun, but none of it makes 42 unique. Its neighbors 41 and 43 belong to plenty of sequences of their own, and mathematicians can play these games with almost any number without learning anything fundamental about the universe. A possible exception, in my opinion, is the Fibonacci sequence, which leads to the golden ratio. Sadly for Adams’ fans, 42 is not a Fibonacci number. The sequence jumps straight from 34 to 55.
The Problem That Almost Crowned 42
One more recent piece of mathematical news put a genuine spotlight on 42.
The “sum of three cubes” problem asks which whole numbers can be written as the sum of three cubes, x³ + y³ + z³, where the cubes may be negative. Some numbers can be ruled out immediately. Any number that leaves a remainder of 4 or 5 when divided by 9 is impossible. For every other number, the only way to settle the question is to find an actual solution, and the solutions can be astronomically large.
By 2009, every eligible number below 100 had been solved except three: 33, 42, and 74. A solution for 74 was found in 2016. In early 2019, Andrew Booker of the University of Bristol found one for 33 after weeks of computing on his university’s supercomputer. That left 42 as the last holdout below 100.
For a brief, glorious moment, it looked as if Adams’ number might earn real mathematical distinction as the stubborn exception. Instead, in September 2019, Booker and Andrew Sutherland of MIT announced that they had cracked it using Charity Engine, a network that borrows idle processing power from hundreds of thousands of home computers around the world. Fittingly, it was the nearest thing yet built to a planet-sized Deep Thought. The answer took more than a million hours of computing rather than seven and a half million years:
42 = (−80,538,738,812,075,974)³ + 80,435,758,145,817,515³ + 12,602,123,297,335,631³
Alas, poor 42 narrowly missed being a number of supreme mathematical significance. It stumped mathematicians and their machines for decades, but not forever. And even if it had held out, “What is the smallest number we cannot yet write as a sum of three cubes?” hardly seems like a compelling candidate for the Ultimate Question.
The Meaning We Bring
Perhaps someday someone will find a way in which 42 is cosmically special in a way that outstrips every other number. I wouldn’t bet on it. Every trail followed above ends the same way. The Egyptian 42s trace back to a map of provinces, the marathon’s 42 to a royal racecourse in 1908, the base-13 trick to a coincidence its author disowned, and the mathematical oddities to properties shared by countless other numbers.
That points to a less flattering conclusion about us than about the number. Faced with a world that does not explain itself, we are pattern-seeking creatures. We grab the nearest thing, give it a significance it has not earned, and then live comfortably with that significance, because it is easier than facing the possibility that the pattern was never there. A number chosen precisely because it meant nothing has become, for millions of people, a symbol of the meaning of everything.
Lest that sound too grim, it is worth remembering that Adams got there first. The joke of Deep Thought was never really the number. It was that the answer is useless if you do not understand the question. “Forty-two will do” as an answer. Finding the right question is still up to us.
